2 Answers
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the AMD64 architecture was positioned by AMD from the beginning as an evolutionary way to add 64-bit computing capabilities to the existing x86 architecture, as opposed to Intel's approach of creating an entirely new 64-bit architecture with IA-64.

The first AMD64-based processor, the Opteron, was released in April 2003.

In fact, in the kernel the 64-bit support is called 'x86_64' to refer to the fact that both AMD and Intel (and others) implement those instructions.

That's not quite correct. I own a Sparc 64-bit CPU made in 1998. The Sparc64 CPUs date back to 1995.
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MarcoOct 31 '12 at 11:56

I see. Edited to reflect the fact that we're talking about x86 CPUs
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RenanOct 31 '12 at 11:57

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Now it's correct. This however raises the question why the packages are not just called x64 or x86-64 instead, in contrast of using a vendor name.
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MarcoOct 31 '12 at 12:00

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@Marco: IIRC x86-64 was name introduced after the Intel released 64-bit processors and x64 was name introduced by Microsoft. Both things happened at the point when some distros already had support for AMD64. I remember the discussion if everything should be renamed to x86-64 (name x64 was not known at that point).
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Maciej PiechotkaOct 31 '12 at 16:56

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Opterons were available before Intel had x86-64 CPUs available and Linux started calling it amd64 to differentiate it from IA-64. Later Intel licensed AMD's 64-bit extensions to include in their own CPUs x86-64 CPUs. Rather than rename everything the name amd64 remained.
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bahamatOct 31 '12 at 20:21

AMD intially implemented x86-64, before Intel. For this reason, most distros (and other OSs such as OpenBSD) did a first release when it was still AMD only, or renaming to x86-64 wasn't considered worth the effort.

I think one of the another contributing factors to this naming confusion was a conflict between Intel and AMD over naming. Intel had an identity crisis in it's 64-bit naming. It cycled through the names EM64T, IA-32e, and finally the official Intel 64. AMD on the other hand always called it AMD64. x86-64 was a fairly obvious name to cover both. However, it wasn't always like this. I remember when it was new and shiny people couldn't figure out what to call it. I've seen these:

x64 -- ambiguous between Itanium and x86-64

IA64 -- again, ambiguous with Itanium

amd64 -- what you see now

EM64T -- rare, but I've seen it before used to refer to both AMD and Intel